游程数的分布和矩的计算

本文较为系统地介绍了游程数的定义、分布及其矩的计算     

The Exact and Limiting Distributions for the Number of Successes in Success Runs Within a Sequence of Markov-Dependent Two-State Trials

The total number of successes in success runs of length greater than or equal to k in a sequence of n two-state trials is a statistic that has been broadly used in statistics and probability. For Bernoulli trials with k equal to one, this statistic has been shown to have binomial and normal distributions as exact and limiting distributions, respectively. For the case of Markov-dependent two-state trials with k greater than one, its exact and limiting distributions have never been considered in the literature. In this article, the finite Markov chain imbedding technique and the invariance principle are used to obtain, in general, the exact and limiting distributions of this statistic under Markov dependence, respectively. Numerical examples are given to illustrate the theoretical results.     

An explicit calculation of the mean of the perimeter of the convex hull of a plane random walk

If (X _n ) _n ⁼¹ is a sequence of i.i.d. random variables in the Euclidean plane such that we compute the mean of the perimeter of theconvex hull ofX ₁++X _k; 0kn}.     

Maximum likelihood estimation of a change-point in the distribution of independent random variables: General multiparameter case

In a sequence ofn independent random variables the pdf changes fromf(x, 0) tof(x, 0 + δv_n⁻¹) after the firstnλ variables. The problem is to estimateλ (0, 1 ), where 0 and δ are unknownd-dim parameters andv_n → ∞ slower thann^1/2. Let_n denote the maximum likelihood estimator (mle) ofλ. Analyzing the local behavior of the likelihood function near the true parameter values it is shown under regularity conditions that ifn_n²(− λ) is bounded in probability asn → ∞, then it converges in law to the timeT_(δjδ)^1/2 at which a two-sided Brownian motion (B.M.) with drift1/2(δ′Jδ)^1/2ton(−∞, ∞) attains its a.s. unique minimum, whereJ denotes the Fisher-information matrix. This generalizes the result for small change in mean of univariate normal random variables obtained by Bhattacharya and Brockwell (1976,Z. Warsch. Verw. Gebiete37, 51–75) who also derived the distribution ofT_μ forμ > 0. For the general case an alternative estimator is constructed by a three-step procedure which is shown to have the above asymptotic distribution. In the important case of multiparameter exponential families, the construction of this estimator is considerably simplified.     

The Maximum Number of Complete Subgraphs of Fixed Size in a Graph with Given Maximum Degree

In this article, we make progress on a question related to one of Galvin that has attracted substantial attention recently. The question is that of determining among all graphs G with n vertices and , which has the most complete subgraphs of size t, for . The conjectured extremal graph is , where with . Gan et al. (Combin Probab Comput 24(3) (2015), 521–527) proved the conjecture when , and also reduced the general conjecture to the case . We prove the conjecture for and also establish a weaker form of the conjecture for all r.     

Joint Distributions of Runs in a Sequence of Multi-State Trials

In this paper we introduce a Markov chain imbeddable vector of multinomial type and a Markov chain imbeddable variable of returnable type and discuss some of their properties. These concepts are extensions of the Markov chain imbeddable random variable of binomial type which was introduced and developed by Koutras and Alexandrou (1995, Ann. Inst. Statist. Math., 47, 743–766). By using the results, we obtain the distributions and the probability generating functions of numbers of occurrences of runs of a specified length based on four different ways of counting in a sequence of multi-state trials. Our results also yield the distribution of the waiting time problems.     

Strong approximation for the general Resten-Spitzer random walk in independent random scenery

This paper is to prove that, if a one-dimensional random walk can be approximated by a Brownian motion, then the related random walk in a general independent scenery can be approximated by a Brownian motion in Brownian scenery.     

Asymptotics for the heat equation in the exterior of a shrinking compact set in the plane via Brownian hitting times

Let and let be a continuous, nonincreasing function on satisfying . Consider the heat equation in the exterior of a time-dependent shrinking disk in the plane:

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If there exist constants and a constant 0$"> such that , for sufficiently large , then . The same result is also shown to hold when is replaced by , where . Also, a discrepancy is noted between the asymptotics for the above forward heat equation and the corresponding backward one. The method used is probabilistic.

     

Recovering a Family of Two-Dimensional Gaussian Variables from the Minimum Process

Suppose that {(X _t, Y _t): t>}0 is a family of two independent Gaussian random variables with means m ₁(t) and m ₂(t) and variances σ ² ₁(t) and σ ² ₂(t). If at every time t>0 the first and second moment of the minimum process X _t∧Y _t are known, are the parameters governing these four moment functions uniquely determined ? We answer this question in the negative for a large class of Gaussian families including the “Brownian” case. Except for some degenerate situation where one variance function dominates the other, in which case the recovery of the parameters is fully successful, the second moment of the minimum process does not provide any additional clues on identifying the parameters.     

Limit Theorems for the Maximum Terms of a Sequence of Random Variables with Marginal Geometric Distributions

Let X _n1 ^* , ... X _nn ^* be a sequence of n independent random variables which have a geometric distribution with the parameter p _n = 1/n, and M _n ^* = \max\{X _n1 ^* , ... X _nn ^* }. Let Z ₁, Z₂, Z₃, ... be a sequence of independent random variables with the uniform distribution over the set N _n = {1, 2, ... n}. For each j N _n let us denote X _nj = min{k : Z_k = j}, M _n = max{X_n1, ... X_nn}, and let S _n be the 2nd largest among X _n1, X_n2, ... X_nn. Using the methodology of verifying D(u_n) and D'(u_n) mixing conditions we prove herein that the maximum M _n has the same type I limiting distribution as the maximum M _n ^* and estimate the rate of convergence. The limiting bivariate distribution of (S_n, M_n) is also obtained. Let _n, _n N_n, , and T _n = min{M(A_n), M(B_n)}. We determine herein the limiting distribution of random variable T _n in the case _n , _n/_n > 0, as n .     

Formulae and Recursions for the Joint Distribution of Success Runs of Several Lengths

Consider a sequence of n independent Bernoulli trials with the j-th trial having probability pj of success, 1 j n. Let M(n,K) and N(n, K) denote, respectively, the r-dimensional random variables (M(n, k1),..., M(n,kr) and (N(n,k1), ..., N(n, kr)), where K = (k1, k2, ..., kr) and M(n, s) [N(n, s)] represents the number of overlapping [non-overlapping] success runs of length s. We obtain exact formulae and recursions for the probability distributions of M(n, K) and N(n, K). The techniques of proof employed include the inclusion-exclusion principle and generating function methodology. Our results have potential applications to statistical tests for randomness.     

Large deviations for Brownian motion in a random scenery

We prove large deviations principles in large time, for the Brownian occupation time in random scenery . The random field is constant on the elements of a partition of ^d into unit cubes. These random constants, say consist of i.i.d. bounded variables, independent of the Brownian motion {B_s,s0}. This model is a time-continuous version of Kesten and Spitzer's random walk in random scenery. We prove large deviations principles in ``quenched' and ``annealed' settings.Mathematics Subject Classification (2000):60F10, 60J55, 60K37     

Exponential Convergence in Probability for Empirical Means of Brownian Motion and of Random Walks

Given a Brownian motion (B _t) _t0 in R ^d and a measurable real function f on R ^d belonging to the Kato class, we show that 1/t ₀ ^t f(B _s ) ds converges to a constant z with an exponential rate in probability if and only if f has a uniform mean z. A similar result is also established in the case of random walks.     

Computational approaches to estimation in the principal component analysis of a stochastic process

After performing a review of the classical procedures for estimation in the principal component analysis (PCA) of a second order stochastic process, two alternative procedures have been developed to approach such estimates. The first is based on the orthogonal projection method and uses cubic interpolating splines when the data are discrete. The second is based on the trapezoidal method. The accuracy of both procedures is tested by simulating approximated sample-functions of the Brownian motion and the Brownian bridge. The real principal factors of these stochastic processes, which can be evaluated directly, are compared with those estimated by means of the two mentioned algorithms. An application for estimation in the PCA of tourism evolution in Spain from real data is also included.     

A sufficient condition and a necessary condition for the diffusion approximations of multiclass queueing networks under priority service disciplines

We establish a sufficient condition for the existence of the (conventional) diffusion approximation for multiclass queueing networks under priority service disciplines. The sufficient condition relates to a sufficient condition for the weak stability of the fluid networks that correspond to the queueing networks under consideration. In addition, we establish a necessary condition for the network to have a continuous diffusion limit; the necessary condition is to require a reflection matrix (of dimension equal to the number of stations) to be completely-S. When applied to some examples, including generalized Jackson networks, single station multiclass queues, first-buffer-first-served re-entrant lines, a two-station Dai–Wang network and a three-station Dumas network, the sufficient condition coincides with the necessary condition.     

On the lifetime of a conditioned Brownian motion in the ball

Consider the Brownian motion conditioned to start in x, to converge to y, with , and to be killed at the boundary ∂Ω. Here Ω is a bounded domain in Rn. For which x and y is the lifetime of this Brownian motion maximal? One would guess for x and y being opposite boundary points and we will show that this holds true for balls in Rn. As a consequence we find the best constant for the positivity preserving property of some elliptic systems and an identity between this constant and a sum of inverse Dirichlet eigenvalues.     

On Number of Occurrences of Success Runs of Specified Length in a Higher-Order Two-State Markov Chain

Let X^-m+1, X^-m+2,..., X⁰, X¹, X²,..., Xⁿ be a time-homogeneous {0, 1}-valued m-th order Markov chain. The probability distributions of numbers of runs of "1" of length k (k m) and of "1" of length k (k < m) in the sequence of a {0, 1}-valued m-th order Markov chain are studied. There are some ways of counting numbers of runs with length k. This paper studies the distributions based on four ways of counting numbers of runs, i.e., the number of non-overlapping runs of length k, the number of runs with length greater than or equal to k, the number of overlapping runs of length k and the number of runs of length exactly k.     

The modulus of non-differentiability of a Brownian motion in l p

Small ball probability is estimated for a Brownian motion in l _p. As an application we establish the modulus of non-differentiability of a Brownian motion in l _p. This revised version was published online in June 2006 with corrections to the Cover Date.